A024912 -id:A024912 - cp's OEIS frontend

A153275 Numbers n such that 10*n+1 is not prime.

0, 2, 5, 8, 9, 11, 12, 14, 16, 17, 20, 22, 23, 26, 29, 30, 32, 34, 35, 36, 37, 38, 39, 41, 44, 45, 47, 48, 50, 51, 53, 55, 56, 58, 59, 61, 62, 65, 67, 68, 71, 72, 73, 74, 77, 78, 79, 80, 83, 84, 85, 86, 87, 89, 90, 92, 93, 95, 96, 98, 100, 101, 104, 107, 108, 110

Offset: 1

Vincenzo Librandi, Dec 22 2008

Multiplied by 2 we have 4,10,16,18,... which is the set of even terms of A153329. - R. J. Mathar, Jan 23 2009

			Distribution of the terms in the following triangular array:
*;
*,*;
2,*,*;
*,*,*,8;
*,*,*,*,12;
*,*,9,*,*,*;
*,*,*,*,*,*,*;
5,*,*,*,*,22,*,*;
*,*,*,17,*,*,*,*,36;
*,*,*,*,23,*,*,*,*,44;
*,*,16,*,*,*,*,39,*,*,*; etc.
where * marks the non-integer values of (2*h*k + k + h)/5 with h >= k >= 1. - _Vincenzo Librandi_, Jan 15 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A024912, A053178, A153329.

[n: n in [0..150] | not IsPrime(10*n + 1)]; // Vincenzo Librandi, Jan 12 2013

q:= n-> not isprime(10*n+1):
select(q, [$0..120])[];  # Alois P. Heinz, Jan 21 2021

Select[Range[0, 200], !PrimeQ[10*# + 1]&] (* Vincenzo Librandi, Jan 12 2013 *)

a(n) = (A053178(n)-1)/10. - R. J. Mathar, Oct 30 2009

63 replaced by 53 by R. J. Mathar, Jan 23 2009

0 added by Arkadiusz Wesolowski, Aug 03 2011

A343717 a(n) is the smallest number that yields a prime when appended to n!.

Original entry on oeis.org

1, 1, 3, 1, 1, 1, 7, 11, 29, 17, 43, 29, 13, 47, 19, 73, 37, 19, 41, 103, 41, 31, 43, 1, 113, 31, 37, 59, 41, 53, 41, 47, 1, 41, 149, 37, 53, 73, 337, 1, 103, 151, 293, 47, 107, 509, 127, 71, 167, 197, 167, 149, 67, 163, 139, 251, 59, 107, 241, 331, 269, 1, 149

Offset: 0

Jon E. Schoenfield, May 17 2021

Appending to n! any number k <= n yields a multiple of k; that multiple cannot be prime except at k=1, so, for every n, a(n)=1 or a(n) > n.
a(n) = 1 iff n = 0 or n is in A024912.
See A068695 for a proof that a(n) always exists. - Felix Fröhlich, May 18 2021
If a(n) is composite, then a(n) > 2n. - Michael S. Branicky, May 18 2021

			n=1: 1! = 1; appending a 1 yields 11, a prime, so a(1)=1.
n=2: 2! = 2; appending a 1 yields 21 = 3*7, and appending a 2 yields 22 = 2*11, but appending a 3 yields 23 (a prime), so a(2)=3.
n=19: 19! = 121645100408832000; appending any number < 103 yields a composite, but 121645100408832000103 is a prime, so a(19)=103.

Lucas A. Brown, Table of n, a(n) for n = 0..2630 (terms 0..1000 from Michael S. Branicky).
Michael S. Branicky, Python program for b-file and A343718, A343719

Cf. A000040, A000142, A024912, A033932, A068695, A095194, A343718, A343719.

a:= proc(n) option remember; local k, t; t:= n!;
      for k while not isprime(parse(cat(t, k))) do od; k
    end:
seq(a(n), n=0..62);  # Alois P. Heinz, May 17 2021

Array[Block[{m = #!, k = 0}, While[! PrimeQ[10^If[k == 0, 1, IntegerLength[k]]*m + k], k++]; k] &, 62] (* Michael De Vlieger, May 17 2021 *)
snp[n_]:=Module[{nf=n!,c=1},While[!PrimeQ[nf*10^IntegerLength[c]+c],c++];c]; Array[snp,70,0] (* Harvey P. Dale, Oct 17 2024 *)

for(n=0,62,my(f=digits(n!));forstep(k=1,oo,2,my(p=fromdigits(concat(f,digits(k))));if(ispseudoprime(p),print1(k,", ");break))) \\ Hugo Pfoertner, May 18 2021

# see link for faster program producing b-file
from sympy import factorial, isprime
def a(n):
  start = str(factorial(n))
  end = 1
  while not isprime(int(start + str(end))): end += 2
  return end
print([a(n) for n in range(63)]) # Michael S. Branicky, May 17 2021

a(n) = A068695(n!) = A068695(A000142(n)).

A102249 Numbers k such that k1111 is prime.

Original entry on oeis.org

10, 13, 31, 36, 51, 57, 61, 69, 72, 78, 79, 90, 91, 97, 117, 120, 127, 129, 135, 138, 153, 156, 166, 174, 183, 184, 189, 201, 205, 210, 222, 225, 226, 234, 237, 240, 241, 244, 252, 261, 265, 273, 276, 280, 285, 292, 304, 306, 309, 318, 322, 325, 327, 337, 345

Offset: 1

Parthasarathy Nambi, Feb 18 2005

			At k=10, k1111 = 101111 (prime).
At k=90, k1111 = 901111 (prime).
At k=138, k1111 = 1381111 (prime).

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A024912.

Select[ Range[ 350], PrimeQ[ FromDigits[ Flatten[ IntegerDigits /@ { #, 1, 1, 1, 1}]]] &] (* Robert G. Wilson v, Feb 21 2005 *)
Select[Range[400],PrimeQ[#*10^4+1111]&] (* Harvey P. Dale, Jul 14 2019 *)

More terms from Robert G. Wilson v, Feb 21 2005

A078656 a(n) = prime(k) where k = n-th prime congruent to 1 mod 10.

Original entry on oeis.org

31, 127, 179, 283, 353, 547, 739, 877, 1087, 1153, 1297, 1523, 1597, 1741, 1823, 2063, 2221, 2749, 2909, 3001, 3259, 3517, 3733, 3911, 4153, 4421, 4663, 4759, 4943, 5189, 5281, 5701, 5801, 6229, 6311, 6841, 7109

Offset: 1

Cino Hilliard, Dec 14 2002

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A024912.

Prime[10Select[Range[110], PrimeQ[10# + 1] &] + 1]
Prime[#]&/@Select[Prime[Range[200]],NumberDigit[#,0]==1&] (* Requires Mathematica version 12 or later *) (* Harvey P. Dale, May 31 2021 *)

pip(n,m,r) = {sr=0; forprime(x=3,n, if(x%m == r,v=prime(x); sr+=1.0/v; print1(v" "); ) ); print(); print("m="m" r="r" sr="sr); }

Edited by Robert G. Wilson v, Dec 17 2002

A089767 Squares which when concatenated with a 1 gives prime.

Original entry on oeis.org

1, 4, 25, 49, 64, 81, 225, 400, 676, 784, 900, 1089, 1225, 1369, 1600, 1681, 2209, 2304, 3249, 3364, 4096, 5041, 6889, 7225, 7396, 8100, 8281, 8649, 9801, 10816, 11025, 11236, 12100, 12544, 12769, 13924, 15876, 16384, 17424, 19881, 21609, 21904

Offset: 1

Amarnath Murthy, Nov 23 2003

Squares n such that 10*n+1 is prime. - Robert Israel, Dec 09 2017

			41, 4001, 6761 etc. are primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A000290 and A024912.

A = []; count = 0; i = 1; while count < 60; s = i*i; if isprime(10*s + 1) A = [A s]; count = count + 1; end; i = i + 1; end; A

select(t -> isprime(10*t+1), map(`^`,[$1..300],2)); # Robert Israel, Dec 09 2017

Select[Range[150]^2, PrimeQ[10 # + 1] &] (* Michael De Vlieger, Dec 09 2017 *)

Corrected and extended by David Wasserman, Feb 25 2004

A102546 Numbers t such that t1 is prime and t is a multiple of 10.

Original entry on oeis.org

10, 40, 60, 70, 120, 130, 160, 180, 190, 280, 300, 330, 370, 400, 420, 480, 510, 550, 570, 580, 610, 630, 670, 700, 790, 810, 850, 900, 960, 990, 1030, 1050, 1060, 1170, 1180, 1210, 1230, 1240, 1260, 1300, 1390, 1440, 1510, 1540, 1560, 1590, 1600, 1630, 1690, 1740, 1830, 1840, 1870

Offset: 1

Parthasarathy Nambi, Feb 24 2005

			If t=10, then t1 = 101 (prime).
If t=180, then t1 = 1801 (prime).
If t=420, then t1 = 4201 (prime).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A024912.

Select[10*Range[200],PrimeQ[10#+1]&] (* Harvey P. Dale, Jan 10 2017 *)

A108857 Numbers n such that 10*n + 131 is prime.

Original entry on oeis.org

0, 2, 5, 6, 8, 11, 12, 14, 15, 18, 20, 27, 29, 30, 33, 36, 39, 41, 44, 47, 50, 51, 53, 56, 57, 62, 63, 68, 69, 75, 78, 81, 84, 86, 89, 90, 92, 93, 96, 102, 104, 105, 107, 110, 116, 117, 119, 123, 125, 132, 134, 135, 138, 140, 144, 147, 149, 159, 161, 167, 168, 170, 173

Offset: 1

Parthasarathy Nambi, Jul 11 2005

			If n=0 then 10*n + 131 = 131 (prime).
If n=33 then 10*n + 131 = 461 (prime).

Matthew House, Table of n, a(n) for n = 1..10000

Cf. A024912.

[n: n in [0..200] | IsPrime(10*n+131)]; // Vincenzo Librandi, Nov 13 2010

select(n -> isprime(10*n+131), [$0..1000]); # Robert Israel, Aug 09 2015

Select[Range[0,200],PrimeQ[10#+131]&] (* Harvey P. Dale, Aug 13 2023 *)

is(n)=isprime(10*n+131) \\ Charles R Greathouse IV, Jun 12 2017

a(n) = A024912(n+6) - 13. - Robert Israel, Aug 09 2015

A243962 Primes p such that 10p + 1, 100p + 1 and 1000p + 1 are also primes.

Original entry on oeis.org

7, 13, 19, 103, 823, 1021, 1579, 1867, 2503, 3331, 5779, 6871, 6949, 9007, 10093, 10399, 11317, 11743, 13327, 13381, 15859, 16657, 17539, 17659, 22189, 26317, 26557, 26821, 27397, 27943, 29209, 29383, 30211, 32443, 37309, 38287, 40213, 40507, 44497, 47569, 47977

Offset: 1

K. D. Bajpai, Jun 16 2014

nonn

			7 is in the sequence because 7 is prime, 10*7 + 1 = 71 is prime, 100*7 + 1 = 701 is prime, and 1000*7 + 1 = 7001 is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A023237, A024912.

with(numtheory):A243962:= proc() local p; p:=ithprime(n); if isprime(10*p+1) and isprime(100*p+1) and isprime(1000*p+1) then RETURN (p); fi; end: seq(A243962 (), n=1..5000);

Select[Prime[Range[10000]], PrimeQ[10 # + 1] && PrimeQ[100 # + 1] && PrimeQ[1000 # + 1] &]

A125869 Numbers k such that p=10k+1 is prime and cos(2Pi/p) is an algebraic number of a 5-smooth degree, but not 3-smooth.

Original entry on oeis.org

1, 3, 4, 6, 10, 15, 18, 24, 25, 27, 40, 54, 60, 64, 75, 81, 120, 160, 162, 180, 216, 225, 300, 400, 405, 480, 486, 648, 768, 810, 864, 900, 960, 972, 1125, 1440, 1536, 1600, 1944, 2160, 2187, 2250, 2304, 2400, 2560, 3240, 3375, 3645, 3750, 4096, 4320

Offset: 1

Artur Jasinski, Dec 13 2006

nonn

Numbers k such that p=10*k+1 is prime and the greatest prime divisor of p-1 is 5.

Cf. A024912, A125866-A125878.

Do[If[Take[FactorInteger[EulerPhi[10n+1]][[ -1]],1]=={5} && PrimeQ[10n+1],Print[n]],{n,1,10000}]

Edited by Don Reble, Apr 24 2007

cp's OEIS Frontend

A153275 Numbers n such that 10*n+1 is not prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A343717 a(n) is the smallest number that yields a prime when appended to n!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A102249 Numbers k such that k1111 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A078656 a(n) = prime(k) where k = n-th prime congruent to 1 mod 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A089767 Squares which when concatenated with a 1 gives prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

Maple

Mathematica

Extensions

A102546 Numbers t such that t1 is prime and t is a multiple of 10.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A108857 Numbers n such that 10*n + 131 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

A125869 Numbers k such that p=10k+1 is prime and cos(2Pi/p) is an algebraic number of a 5-smooth degree, but not 3-smooth.